Meriam, Kraige & Bolton — Rigid Bodies: Equivalent Systems of Forces: Problem 3_23

⚡ Mecademy AIENG정역학 · ch3  Problem Statement Determine the moment which the motor must exert in order to position the uniform slender bar of mass and length in the arbitrary position . The ratio of the radius of the gear wheel attached to the bar to that of the gear wheel attached to the motor shaft is 2. Problem 3/23 (a) Determination of the motor moment 1. Present Final Formula: The equilibrium of the bar requires that the moment exerted by gear at the pivot must balance the moment due to the weight of the bar. The motor torque on gear is related to the torque on gear by the ratio of their radii. 2. Substitute Values: The mass of the uniform bar is , and its center of gravity is located at a distance of from the pivot . The moment arm of the weight relative to is . From the problem statement, the radius ratio is given as: Substituting these into the torque relationship: 3. Partial Operations: Calculate the combined coefficients for the moment: 4. Final Calculation: The final expression for the motor moment required to maintain equilibrium at an angle is: M mLθ BA M B MAB M = B m g sin θ 2 L M=M B ( r B r A ) m L/2B B sinθ 2 L = r A r B 2⟹ = r B r A 2 1 M=mg⋅ sinθ ⋅ ( 2 L ) 2 1 M= ⋅ mgLsin θ( 2 1 2 1 ) M= mgLsin θ 4 1 M θ ● Final Conclusion: The motor must exert a moment to maintain the bar in equilibrium at the specified position. This moment counteracts the gravitational torque with a mechanical advantage of 2 provided by the gear system. ✨ Final Answer Summary (a) Mecade